Abstract
The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector in .
1. Introduction
Kinematically, a sweeping surface is a surface traced by a one-parameter family of spheres with centers on a regular space curve, its directrix or spine. If the radii of the spheres are fixed, the sweeping surface is called tubular. There are several examples that we are familiar with, such as circular cylinder (spine is a line, the axis of the cylinder), right circular cone (spine is a line (the axis), radii of the spheres not constant), torus (directrix is a circle), and rotation surface (spine is a line). This visualization is a popularization of the classical notation of a partner of a planar curve [1–4]. One of the noteworthy facts linked with the sweeping surface is that the sweeping surface can be developable surface, that is, can be developed onto a plane without tearing and stretching. Therefore, sweeping surfaces have great usefulness in considerable product design which uses leather, paper, and sheet metal as materials (see, e.g., [5–8]). The developable surface can be represented using the Serret–Frenet frame of space curves from the viewpoint of singularity theory. In [9], Izumiya and Takeuchi defined the rectifying developable surfaces of space curves, where they proved that a regular curve is a geodesic of its rectifying developable surface and revealed the relationship between singularities of the rectifying developable surface and geometric invariants. Ishikawa investigated the relationship between the singularities of tangent developable surfaces and some types of space curves [10]. He also gave a classification of tangent developable surfaces by using the local topological property. There are several works about the singularity theory of developable ruled surfaces by using the Serret–Frenet frame of space curves, for example, [11–16]. However, the Serret–Frenet frame is undefined wherever the curvature vanishes, such as at points of inflection or along straight sections of the curve. A new frame is needed for the kind of mathematical analysis that is typically done with computer graphics. Therefore, Bishop [17] introduced the rotation minimizing frame (RMF) or Bishop frame, which could provide the desired means to ride along a space curve with vanished second 1derivative. After that, many research works linked to the RMF have been treated in the Euclidean space and Minkowski space [18–23].
In this paper, the classification of singularity type of timelike sweeping surfaces is studied with the RMF in . We present a new invariant related to the singularities of these sweeping surfaces. It is demonstrated that the generic singularities of this sweeping surface are cuspidal edge and swallowtail, and the types of these singularities can be characterized by this invariant, respectively. Afterwards, we have solved the problem of requiring the surface that is timelike sweeping surface and at the same time spacelike/timelike developable surface. Two examples are presented to explain the theoretical results.
2. Preliminaries
In this section, we give some definitions and basic concepts that we will use in this paper (see, for instance, [1, 8, 24]). Let be a 3-dimensional Cartesian space. For any and , the pseudoscalar product of and is defined by
We call Minkowski 3-space. We write instead of . We say that a nonzero vector is spacelike, lightlike, or timelike if , , or , respectively. The norm of the vector is defined to be . For any two vectors , , we define the cross product by where , , and are the canonical basis of . The hyperbolic and Lorentzian unit spheres, respectively, are
Let be a unit speed spacelike curve with timelike binormal normal in ; by and , we denote the natural curvature and torsion, respectively. Consider the Serret–Frenet frame { , } associated with curve , then the Serret–Frenet formulae read where , , and are called the unit tangent vector, the principal normal vector, and the binormal vector, respectively. Here, “prime” denotes the derivative with respect to the parameter . The Serret–Frenet vector fields satisfy the relations
The Bishop frame or rotation minimizing frame (RMF) of is defined by the alternative frame equations where is RMF Darboux vector. Here, the Bishop curvatures are defined by , . The relation matrix can be expressed as where is a hyperbolic angle. One can show that
We define a Bishop spherical Darboux image as
Then, we define a new geometric invariant .
A ruled surface in is locally the map : defined by where is called the directrix curve and the director curve. The straight lines are called rulings. It is well known that is a developable surface iff det.
Definition 1. A surface in the Minkowski 3-space is called a timelike surface if the induced metric on the surface is a Lorentz metric and is called a spacelike surface if the induced metric on the surface is a positive definite Riemannian metric, i.e., the normal vector on spacelike (timelike) surface is a timelike (spacelike) vector.
3. Timelike Sweeping Surfaces and Singularities
In this section, the classification of singularity type of timelike sweeping surfaces is studied with the RMF in . Let be a unit speed spacelike curve with timelike binormal as defined on the RMF frame. Then, we can give the parametric form of sweeping surface given by the spine curve as follows: where is called planar profile (cross section); “” represents transposition, with another parameter . The semiorthogonal matrix specifies the RMF along . We will utilize “dot” to indicate the derivative with respect to the arc length parameter of the profile curve .
The tangent vectors and the unit normal vector to the surface, respectively, are
From Equation (3.3), it follows that is contained in the normal plane of the spine curve , since it is orthogonal to . Thus, the normal of the profile curve and the surface normal are identical. Through this work, we will assume that the profile curve is a unit speed timelike curve, that is, . Thus, is a timelike sweeping surface. From now on, we shall often not write the parameter explicitly in our formulae.
Our aim of this work is the following theorem.
Theorem 2. For the timelike sweeping surface Equation (3.1), with , one has the following
(A)
(1) is locally diffeomorphic to a timelike line {} at iff (2) is locally diffeomorphic to the cusp at iff and (B)
(1) is locally diffeomorphic to cuspidal edge at iff and (2) is locally diffeomorphic to swallowtail SW at iff , , and Here, , , and . The graphs of , , and are seen in Figures 1–3.
3.1. Lorentzian Bishop Height Functions
Now, we will define two different families of Lorentzian Bishop height functions that will be useful to study the singularities of as follows: , by . We call it the Lorentzian Bishop height function. We use the notation for any fixed . We also define , by . We call it the extended Lorentzian Bishop height function of . We denote that . From now on, we shall often not write the parameter . Then, we have the following proposition.
Proposition 3. Let be a unit speed spacelike curve with timelike binormal normal and . Then, the following hold:
(A)
(1) iff and (2) iff (3) iff and (4) iff and (5) iff and (B)
(1) iff there exist (2) iff there exist such that and (3) iff , , and (4) iff , , and (5) iff , , and (6) iff , , and
Proof. According to Equation (2.2), we have iff .
(A)
(1)Since and { , } is RMF along , then there exist , such that . By the condition that , we get . The converse direction also holds(2)Since , we have . It follows from the fact that and . Therefore, we haveTherefore, iff .
(3)Since , by the conditions of (2), we haveThus, iff and .
(4)Sinceby the conditions of (7), we have
Thus, iff and .
(5)Since , we haveBy using the conditions of (8), we have
Therefore, iff and .
(B)
Using the same computation as the proof of (A), we can get (B) (4) .☐
Proposition 4. Let be a unit speed spacelike curve with timelike binormal and . Then, we have iff is a constant timelike vector.
Proof. By simple calculations, we have Thus, iff ☐
Proposition 5. Let : be a unit speed spacelike curve with timelike binormal and . Then, we have the following. (a) is a B-slant helix iff is constant(b) is a part of circle on whose center is the timelike constant vector
Proof. (a)Suppose that . Hence, we can writeThus, , that is, is a B-slant helix. (b)SinceThis means that is a part of circle on whose center is the constant timelike vector .☐
3.2. Unfolding of Functions by One Variable
In this subsection, we use some general results on the singularity theory for families of function germs. Let be a smooth function and . Then, is called an -parameter unfolding of . We say that has -gularity at if for all and . We also say that has A-gularity () at s. Let the -jet of the partial derivative at be (without the constant term), for . Then, is called a -versal unfolding if the matrix of coefficients has rank . So, we write important set about the unfolding relative to the above notations. The discriminant set of is the set
The bifurcation set of is the set
We can also give the following theorem [12, 13].
Theorem 6. Let be an -parameter unfolding of , which has the singularity at .
Suppose that is a -versal unfolding.
(a)If , then is locally diffeomorphic to and (b)If , then is locally diffeomorphic to , and is locally diffeomorphic to (c)If , then is locally diffeomorphic to , and is locally diffeomorphic to Hence, we have the following fundamental proposition.
Proposition 7. Let : be a unit speed spacelike curve with timelike binormal and . (1) If has an -singularity at , then is a -versal unfolding of . (2) If has an -singularity at , then is a -versal unfolding of
Proof. (see (4)).
Since , , , , and cannot be all zero. Without loss of generality, we may assume that . Then, by , we have
Thus, we have
Therefore, the 2-jets of at () are as follows. Let , and assume that , then
(i)If has the -singularity at , then . So the matrix of coefficients isSuppose that the rank of the matrix is zero, then we have
Since , we have so that we have the contradiction as follows:
Therefore, rank , and is the () versal unfolding of at .
(ii)If has the -singularity at , then , and by Proposition 3.where , , and . So, the matrix of the coefficients is
For the purpose, we also require the matrix to be nonsingular, which always does. In fact, the determinate of this matrix at is
Since , we have . Substituting these relations to the above equality, we have
This means that rank .
(2) Under the same notations as in (4), we have
We require the matrix
to have the maximal rank. By case (4) in Equation (3.14), the second row of does not vanish, so rank .☐
Proof of Theorem 1 (see (4)). By Proposition 3, the bifurcation set of is
The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6. The discriminant set of is given as follows:
The assertion (4) of Theorem 2 follows from Propositions 3 and 7 and Theorem 6.
Example 1. Given the spacelike helix It is easy to show that Then, . If we choose , for example, we have We can calculate the geometric invariant We also have The timelike Bishop spherical Darboux image is shown in Figure 4) The timelike sweeping surface family is By choosing and , then we immediately have a timelike sweeping surface (see Figure 5).
3.3. Developable Surfaces
Developable surfaces can be briefly introduced as special cases of ruled surfaces. Such surfaces are widely used, for example, in the manufacture of automobile body parts, airplane wings, and ship hulls. Therefore, we analyze the case that the profile curve degenerates into a timelike line. Then, we have the following timelike developable surface
We also have the following spacelike developable surface
It is clear that (resp. ), , that is, the surface (resp. ) interpolate the curve . Also, we have
Thus, we have that (resp. ) is nonsingular at if and only if (resp. ). We designate to represent (), and based on Theorem 3.3 in [23], we can give the following corollary.
Corollary 8. For the developable ruled surfaces and , we have the following: (1) (resp. ) is locally diffeomorphic to the cuspidal edge CE at iff and (2) (resp. ) is locally diffeomorphic to swallowtail SW at iff and
Example 2. By making using of Example 1, we have the following: (1)If , then and . The timelike developable surfaceis locally diffeomorphic to the cuspidal edge; see Figure 6. We can obtain the singular locus of as follows: (2)If , then and . The spacelike developable surfaceis locally diffeomorphic to swallowtail; see Figure 7. Also, the singular locus of is
4. Conclusion
In this paper, we introduced the notion of timelike sweeping surfaces with rotation minimizing frames in Minkowski 3-space . By applying singularity, we classified the generic properties and present a new geometric invariant related to the singularities of this timelike sweeping surface. It leads to the fact that the generic singularities of this sweeping surface are cuspidal edge and swallowtail, and the types of these singularities can be characterized by this geometric invariant, respectively. Finally, some examples are presented to explain the theoretical results.
Data Availability
All of the data are available within the paper.
Conflicts of Interest
The authors declare that they have no conflict of interest.